Suppose we have a right triangle with integer sides

**NikoBellic** · April 19th 2010, 06:21 PM

Suppose we have a right triangle with integer sides (in some unit of measure). Prove
(a) One of the legs has a length divisible by 3.
(b) One of the three sides has length divisible by 5.

**chiph588@** · April 19th 2010, 06:39 PM

Originally Posted by NikoBellic

Suppose we have a right triangle with integer sides (in some unit of measure). Prove
(a) One of the legs has a length divisible by 3.

We want to show either $m^2-n^2$ or $2mn$ is divisible by $3$ .

WLOG if $m\equiv0\bmod{3}$ then we're done since then $2mn\equiv0\bmod{3}$ .

So assume $m$ and $n$ are both not divisible by $3$ .

But then by flt we get that $m^2\equiv1\bmod{3}$ and $n^2\equiv1\bmod{3}$ , so $m^2-n^2\equiv0\bmod{3}$ .

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